Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the philosophical treatment of mathematics, necessity, or both, although there is no single thread running through all three essays.

Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.

Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]

Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.

Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.

Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.

According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.

This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the...

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